Microsoft Word - Cengel and Boles TOC _2-03-05_.doc

which yields

(b)

where we have neglected the second-order term dV^2. The amplitude of the

ordinary sonic wave is very small and does not cause any appreciable

change in the pressure and temperature of the fluid. Therefore, the propaga-

tion of a sonic wave is not only adiabatic but also very nearly isentropic.

Then the second Tdsrelation developed in Chapter 7 reduces to

or

(c)

Combining Eqs. a, b, and c yields the desired expression for the speed of

sound as

or

(17–9)

It is left as an exercise for the reader to show, by using thermodynamic

property relations (see Chap. 12) that Eq. 17–9 can also be written as

(17–10)

where kis the specific heat ratio of the fluid. Note that the speed of sound

in a fluid is a function of the thermodynamic properties of that fluid.

When the fluid is an ideal gas (P
rRT), the differentiation in Eq. 17–10

can easily be performed to yield

or

(17–11)

Noting that the gas constant Rhas a fixed value for a specified ideal gas and

the specific heat ratio kof an ideal gas is, at most, a function of tempera-

ture, we see that the speed of sound in a specified ideal gas is a function of

temperature alone (Fig. 17–9).

A second important parameter in the analysis of compressible fluid flow

is the Mach numberMa, named after the Austrian physicist Ernst Mach

(1838–1916). It is the ratio of the actual velocity of the fluid (or an object in

still air) to the speed of sound in the same fluid at the same state:

(17–12)

Note that the Mach number depends on the speed of sound, which depends

on the state of the fluid. Therefore, the Mach number of an aircraft cruising

Ma

V

c

c
 2 kRT

c^2 
ka

0 P

0 r

b

T

kc

01 rRT 2

0 r

d

T

kRT

c^2 
ka

0 P

0 r

b

T

c^2 
a

0 P

0 r

b

s

c^2 

dP

dr

¬¬at s
constant

dh

dP

r

T ds 
dh

dP

r

dhc dV
 0

828 | Thermodynamics

AIR HELIUM

347 m/s

634 m/s

200 K

300 K

1000 K

284 m/s

1861 m/s

1019 m/s

832 m/s

FIGURE 17–9

The speed of sound changes with

temperature and varies with the fluid.

0

¡

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